Thermal Buckling and Postbuckling of Euler-Bernoulli Beams Supported on Nonlinear Elastic Foundations

Abstract

Deformations of homogeneous and isotropic pinned-pinned and fixed-fixed Euler-Bernoulli beams supported on nonlinear elastic foundations and heated uniformly into the postbuckling regime have been analyzed numerically. Geometric nonlinearities introduced by finite deflections and curvature of the deformed beams are incorporated in the problem formulation. First buckling due to the uniform temperature rise and buckling mode transitions are investigated analytically by analyzing the linear problem. Subsequently, the nonlinear boundary-value problems for postbuckling of beams are transformed into initial-value problems and analyzed by the shooting method.

Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation

“…We assume that the reaction force exerted on the beam by the non-linear elastic foundation can be expressed in terms of the displacements of beam's centroidal axis as [7] …”

“…(25)- (28) and (30), i.e., setting 0 = 1, sin θ ≈ θ ≈ W ′ , cos θ = 1, and P H = φ 4 τ , and neglecting all non-linear terms in the unknown functions, we get a linear equation for the critical buckling temperature of a FGM beam resting on the elastic foundation; e.g., see Ref. [7]. In the numerical work, it can be found in the limit of m 0 (or ϕ 0 ) approaching zero.…”

“…That is, a higher-order buckling mode could have a lower buckling temperature than that for the immediately lower order buckling mode. The coordinates of the transition points in the buckling modes are listed in Tables 2 and 3 and are compared with the results of Li and Batra [7], and of Wu and Zhong [24] for homogeneous EBBs. Wu and Zhong studied a mechanical problem, thus values of τ cr for the mode transition are not listed in the table.…”

“…(K 1 /π 4 , τ cr /π 2 ) [7]; **Results from [24]. mid-section, the de ection of the beam centroid can also be regarded as the continuation parameter for an odd order buckling mode, such as the rst, the third, and the h.…”

“…Thermal buckling and post-buckling deformations of Euler-Bernoulli beams (EBBs) resting on twoparameter non-linear elastic foundations were studied by Song and Li [6] and Li and Batra [7] for di erent boundary conditions. It was found that the critical temperature depended linearly on the linear elastic foundation sti ness parameter.…”